** R ELIABILITY A LLOCATION : T HEORY**

**5.4. Related works: classical approaches**

This section presents a systematic review of the most common RA methods available in literature.

### 5.4.1. Equal method

The simplest and easiest allocation method is the "Equal Reliability Allocation".

As it can be easily guessed from the name, this method allocates the same failure rate and the same reliability to all the components making up the

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system. This means that the weight factor πππ£π£ assessed using the Equal method is the same for all components i. As a consequence, the Equal method could be applied only to provide a first rough estimation of the reliability values to be allocated, but it cannot be considered a valuable solution.

The mathematical model of the equal allocation method is the following:

π
π
π£π£β(π‘π‘) = οΏ½π
π
^{ππ} ππππππβ(π‘π‘)= [π
π
ππππππβ(π‘π‘)]^{ππ}^{1} (5.19)
πππ£π£= 1

π π (5.20)

### 5.4.2. ARINC method

The ARINC apportionment method was designed in 1964 by ARINC Research Corporation, a subsidiary of Aeronautical Radio, Inc [157].

This method is based on the assumption that the reliability of components can be assessed using previous calculations on similar components.

The mathematical expression of weight factors is the following:

πππ£π£= ππ_{π£π£}

ππππππππ= ππ_{π£π£}

β^{ππ}_{ππ=1}ππ_{ππ} (5.21)

Where πππ£π£ is the estimated failure rate of the component i-th obtained through a similar system and ππππππππ is the estimated failure rate of the whole architecture [157].

The peculiarity of the ARINC technique is that it is one of the few methods that considers historical failure data to assess the weight factors rather than quantitative influence factors like most of other techniques. As a matter of fact, ARINC requires the knowledge of past allocations on similar systems to allocate reliability to the various levels of the current system.

The main advantage of this method is essentially its simplicity of calculations which allows to rapidly implement the allocation. However, ARINC suffers many flaws, such as:

β’ It is not possible to apply ARINC method to innovative systems since no past data related to a similar system are available.

β’ All failure rates must be extracted from the same source (single database), as they must be comparable to each other in order to have an optimal allocation.

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### 5.4.3. AGREE method

AGREE (Advisory Group on Reliability of Electronic Equipment) technique considers three influence factors to calculate the weighting factors of each subsystem [158]. Complexity πΆπΆπ£π£ is assessed as the number of elements of the generic subsystem πππ£π£ compared to the total number of components π π π‘π‘π π π‘π‘ of overall configuration.

πΆπΆπ£π£= πππ£π£

π π ππππππ (5.22)

This technique also considers the importance πΌπΌπ£π£ of each subsystem i, where importance is defined as the probability that the system fails when the subsystem fails, thus πΌπΌπ£π£β [0; 1] where πΌπΌ = 1 stands for the most critical items, while πΌπΌ = 0 means that the failure has no critical effects.

The third factor takes into account the effective time of use π‘π‘π£π£β² of the subsystems, as follow:

π‘π‘_{π£π£}^{β²}= π‘π‘

π‘π‘π£π£ (5.23)

where π‘π‘π£π£ is the time of use of item I, while π‘π‘ is the time of use of the whole system.

According to the AGREE method [158], the reliability of a series architecture composed by N subsystems is defined as follow:

π π ππππππβ(π‘π‘) = οΏ½{1 β πΌπΌπ£π£[1 β π π π£π£β(π‘π‘π£π£)]}

ππ π£π£=1

= οΏ½οΏ½1 β πΌπΌπ£π£οΏ½1 β ππ^{βππ}^{ππ}^{β}^{π‘π‘}^{ππ}οΏ½οΏ½

ππ π£π£=1

(5.24)

Using the Taylor approximation of the exponential function ππ^{βπ₯π₯} β 1 β π₯π₯ when
π₯π₯ β 0, then:

π π ππππππβ(π‘π‘) β οΏ½{1 β πΌπΌπ£π£[1 β (1 β πππ£π£βπ‘π‘π£π£)]}

ππ π£π£=1

= οΏ½{1 β πΌπΌπ£π£πππ£π£βπ‘π‘π£π£}

ππ π£π£=1

(5.25)

Introducing the Taylor approximation once again and rewriting the system reliability as exponential function:

π
π
ππππππβ(π‘π‘) β οΏ½οΏ½ππ^{βπΌπΌ}^{ππ}^{ππ}^{ππ}^{β}^{π‘π‘}^{ππ} οΏ½

ππ π£π£=1

= ππ^{β β}^{ππππππππ}^{ππ=1} ^{οΏ½πΌπΌ}^{ππ}^{ππ}^{ππ}^{β}^{π‘π‘}^{ππ}^{οΏ½} (5.26)

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To solve equation (5.26) it is necessary to rewrite the left term using the exponential function.

Thus, considering the properties of exponential and logarithmic functions, equation (5.26) can be rewritten as follow:

ππ^{π π ππ[π
π
}^{ππππππ}^{β}^{(π‘π‘)]}= ππ^{β β}^{ππ}^{ππ=1}^{οΏ½πΌπΌ}^{ππ}^{ππ}^{ππ}^{β}^{π‘π‘}^{ππ}^{οΏ½} (5.27)
ππππ[π
π
ππππππβ(π‘π‘)] = β οΏ½ πΌπΌπ£π£πππ£π£βπ‘π‘π£π£

ππ π£π£=1

(5.28)

Multiplying and dividing the first term of equation (5.28) by the same quantity π π ππππππ:

π π ππππππ

π
π
_{ππππππ} ππππ[π
π
_{ππππππ}^{β}(π‘π‘)] = β οΏ½ πΌπΌπ£π£ππ_{π£π£}^{β}π‘π‘_{π£π£}

ππ π£π£=1

(5.29)

However, π π ππππππ is the total number of components that make up the entire system. Thus, considering the definition of Complexity introduced by the AGREE method in equation (5.22), π π ππππππ can be rewritten as follow:

π
π
_{ππππππ}= οΏ½ ππ_{π£π£}

ππ π£π£=1

(5.30)

Introducing equation (5.30) within equation (5.29):

οΏ½ πΌπΌπ£π£πππ£π£βπ‘π‘π£π£ ππ π£π£=1

= βππππ[π
π
_{ππππππ}^{β}(π‘π‘)] β β ππππ π£π£
π£π£=1

π π ππππππ (5.31)

Then, using the properties of the summation:

οΏ½ πΌπΌπ£π£πππ£π£βπ‘π‘π£π£ ππ π£π£=1

= β οΏ½ οΏ½πππ£π£βππππ[π π ππππππβ(π‘π‘)]

π π ππππππ οΏ½

ππ π£π£=1

(5.32)

πΌπΌπ£π£πππ£π£βπ‘π‘π£π£= βπππ£π£βππππ[π π ππππππβ(π‘π‘)]

π
π
_{ππππππ} (5.33)

91 Introducing the definition of Complexity πΆπΆπ£π£ as in equation (5.22) and the definition of effective time as in equation (5.23) the latter became:

ππ_{π£π£}^{β}= βπΆπΆπ£π£β π‘π‘_{π£π£}^{β²}β ππππ[π
π
ππππππβ(π‘π‘)]

πΌπΌ_{π£π£}π‘π‘ (5.34)

Now it is possible to define the weight factor of the AGREE method as a function of complexity, importance and effective time:

ππ_{π£π£}=πΆπΆπ£π£β π‘π‘_{π£π£}^{β²}

πΌπΌπ£π£ (5.35)

Introducing equation (5.35) and equation (5.2) within equation (5.34) the allocated failure rate according to the AGREE method could be expressed as:

πππ£π£β= βπππ£π£β ππππ[π π ππππππβ(π‘π‘)]

π‘π‘ = πππ£π£β ππππππππβ (5.36)

The AGREE technique is a milestone in RA approaches. However, it suffers major drawbacks, such as:

β’ The importance factor, as it is defined, does not take into account the consequences that a subsystem failure induced on the system.

β’ It requires Taylor approximation, thus obtaining approximate result.

β’ The assessment of the weight factor takes into account only three influence factors.

### 5.4.4. FOO method

The FOO (Feasibility-Of-Objectives) technique was first introduced in 1976 by Anderson [159] and then included into the MIL-HDBK-338B Electronic Reliability Design Handbook from Department of Defense of USA in 1988 [148]

as a method to develop and implement reliability programs for generic military
products. Following the FOO method, the subsystem allocation factors are
**computed as a function of four influence factors, namely complexity C, **
**environmental factor E, state of the art A and operative time O. Each rank is **
estimated using both design engineering and expert judgments and it is based
on a scale from 1 to 10 as detailed described in TABLE V.II.

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TABLE V.II

RULES FOR THE ASSESSMENT OF INFLUENCE FACTORS ACCORDING TO FOO METHOD.
**I****NFLUENCE ****F****ACTORS** **R****ATING**

COMPLEXITY -**C ** **1**23456789**10 **
**L****OW ****M****AX**
ENVIRONMENT CONDITION -**E ** **1**23456789**10 **

**L****OW ****M****AX**
STATE OF THE ART -**A ** **1**23456789**10 **

**M****AX ****L****OW**
OPERATING TIME -**T ** **1**23456789**10 **

**M****AX ****L****OW**

The rating values are then multiplied to achieve a partial weight factor π½π½π£π£.

π½π½π£π£= πΆπΆπ£π£β πΈπΈπ£π£β π΄π΄π£π£β πππ£π£ (5.37) The final product has values ranging from 1 to 10000 and the subsystem ratings are normalized so that their sum is equal to 1.

Thus, the weighting factors are given by:

πππ£π£= πΆπΆ_{π£π£}β πΈπΈ_{π£π£}β π΄π΄_{π£π£}β ππ_{π£π£}

β οΏ½πΆπΆ^{ππ}_{ππ=1} ππβ πΈπΈ_{ππ}β π΄π΄_{ππ}β ππ_{ππ}οΏ½ =_{β}^{π½π½}^{π£π£}_{π½π½}

ππ π£π£

ππ=1 (5.38)

The FOO method is a simple technique easily implementable using software tools. However, it is characterized by some major flaws (quite similar to the RPN drawbacks described in Section 3.2):

β’ The partial weight factor π½π½π£π£ is not unique. In fact, different combinations of the influence factors could provide the same π½π½π£π£.

β’ Although the partial weight factor π½π½π£π£ could assume values between 1 and 10000, there are many gaps in the range and only a very limited part of these 10000 possible values is obtained from a unique combination of factors.

β’ All the different combinations of influence factors that lead to the same partial weight factor will also lead to the same allocated reliability. This may not be correct as the nature of the influence factors producing the same π½π½π£π£ can be remarkably different.

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β’ All the four influence factors have the same importance within the equation.

β’ High subjectivity of the definition, which is deeply influenced by the expertβs judgments.

### 5.4.5. Bracha method

Bracha method uses the same factors of FOO technique (see TABLE V.II) but
**it privileges the state of the art factor A in the formula to calculate the partial **
weight factors π½π½π£π£ [160]:

π½π½π£π£= π΄π΄π£π£β (πΆπΆπ£π£+ πππ£π£+ πΈπΈπ£π£) (5.39) According to the Bracha method, the values of the influence factors are not determined by an expert like the FOO approach. Instead, the influence factor ratings are calculated through a set of complex mathematical models using several base factors, some of them are listed below:

β’ the number of components of each subsystem;

β’ the number of components of the most complex subsystem;

β’ the number of redundancies;

β’ the time of use of each subunit;

β’ the operating time of each subsystem;

β’ the applied stress;

β’ the age of the database;

β’ the time required to design the system.

These models result in a set of four influence factors mathematically estimated varying in the range from 0 to 1. The subsystem ratings are then normalized, therefore the weighting factors are given by [160]:

πππ£π£= π΄π΄π£π£β (πΆπΆπ£π£+ πππ£π£+ πΈπΈπ£π£)

β [π΄π΄^{ππ}_{ππ=1} π£π£β (πΆπΆπ£π£+ πππ£π£+ πΈπΈπ£π£)] =_{β}^{π½π½}^{π£π£}_{π½π½}

ππ π£π£

ππ=1 (5.40)

The Bracha method is able to solve two out of five drawbacks of the FOO method, namely the high subjectivity of the factor definition and the same importance assigned to all the factor in the equation to calculate πππ£π£. However, it is not able to solve the other three major drawbacks of the FOO method, and it is also characterized by a high computational complexity due to the

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**models required to assess the complexity C, the environmental factor E, the **
**state of the art A and the operative time O. **

### 5.4.6. Karmiol method

The Karmiol method is based on the assessment of four influence factors,
**namely complexity C, state of the art A, operative time O and Criticality K. **

Each rank is estimated using both design engineering and expert judgments and it is based on a scale from 1 to 10 [161].

The procedure used to calculate the partial weight factor π½π½π£π£ and the weight factor πππ£π£ is quite similar to the FOO model. The only difference is that the Karmiol method allows two different approaches. In the first one the partial weight factor π½π½π£π£ is based on the product of the indexes similarly to the FOO, as follow:

π½π½_{π£π£}= πΆπΆ_{π£π£}β π΄π΄_{π£π£}β ππ_{π£π£}β πΎπΎ_{π£π£} (5.41)
Then, the weight factor is achieved after a normalization process to ensure that
equation (5.10) is satisfied. Thus:

πππ£π£= πΆπΆ_{π£π£}β π΄π΄_{π£π£}β ππ_{π£π£}β πΎπΎ_{π£π£}

β (πΆπΆ^{ππ}_{ππ=1} π£π£β π΄π΄_{π£π£}β ππ_{π£π£}β πΎπΎ_{π£π£} ) =_{β}^{π½π½}^{π£π£}_{π½π½}

ππ π£π£

ππ=1 (5.42)

Alternatively, it is possible to calculate the partial weight factor as sum of the indexes and then evaluate the weight factor after the normalization process, as follow:

π½π½π£π£= πΆπΆπ£π£+ π΄π΄π£π£+ πππ£π£+ πΎπΎπ£π£ (5.43)
ππ_{π£π£}= πΆπΆπ£π£+ π΄π΄π£π£+ πππ£π£+ πΎπΎπ£π£

β (πΆπΆ^{ππ}_{ππ=1} π£π£+ π΄π΄π£π£+ πππ£π£+ πΎπΎπ£π£ ) =_{β}^{π½π½}^{π£π£}_{π½π½}

ππ π£π£

ππ=1 (5.44)

### 5.4.7. AWM method

In 1999 Kuo [162] introduced an Averaging Weighted Method (AWM) as a guide for reliability allocation design.

The method uses a questionnaire investigation to select the most influential system reliability factors such as complexity, state-of-the-art, system criticality, environment, safety, and maintenance in order to determine the subsystem reliability allocation ratings. All the influence factors included in Fig. 5.3 are

95 allowed, depending on the results of the questionnaire. Each rank is estimated on a scale from 1 to 10 using design engineering and expert judgments to obtain the subsystem reliability rate [162]. TABLE V.III shows the admissible influence factors and their rating rules according to the guidelines described in section 5.3.2.

TABLE V.III

INFLUENCE FACTORS ADMISSIBLE BY AWM ALLOCATION METHOD

**I****NFLUENCE **

**F****ACTORS** **D****ESCRIPTION** **R****ATING**

COMPLEXITY -**C ** Number of components;

system architecture. **1**23456789**10 **
**L****OW ****M****AX**
ENVIRONMENT

CONDITION -**E **

External stress factors (humidity, temperature, vibration, etc.).

**1**23456789**10 **
**L****OW ****M****AX**
STATE OF THE ART

**-A **

Scientific development in the system specific engineering context.

**1**23456789**10 **
**M****AX ****L****OW**
CRITICALITY -**K ** Subsystem importance;

consequences of a potential fault on the entire system.

**1**23456789**10 **
**M****AX ****L****OW**
MAINTAINABILITY -**M **Average repair cost; average

repair time. **1**23456789**10 **
**L****OW ****M****AX**
SAFETY -**R ** Impact of failure on system

safety **1**23456789**10 **

**M****AX ****L****OW**

Considering a system composed by N subsystem, m is the number of influence
factors and p the number of experts. Let πππ£π£ππ denotes the j-th rating for
subsystem i. ππ_{πΎπΎπ£π£ππ} is the j-th rating for subsystem i set by L-th expert and each
factor is defined as follows:

πππ£π£ππ =1

πποΏ½ ππ_{πΎπΎπ£π£ππ}

ππ ππ=1

βππ = 1, β¦ , ππβππ = 1, β¦ , π π (5.45)

Then, similarly to the Karmiol method, also in this case two different models can be used to allocate weighting factors πππ£π£.

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The geometric model is based on the product of the influence factors, and thus the weight factor is given by:

πππ£π£= β^{ππ}_{ππ=1}ππ_{π£π£ππ}

β βππ πππ£π£ππ ππ ππ=1

ππ=1 = π½π½ππ

βππ π½π½ππ

ππ=1 (5.46)

While the arithmetic model is based on the sum of the influence factors:

πππ£π£= βππ πππ£π£ππ ππ=1

β βππ πππ£π£ππ ππ ππ=1

ππ=1 = π½π½_{ππ}

β^{ππ}_{ππ=1}π½π½_{ππ} (5.47)

The AWM has many advantages, such as:

β’ The higher the experts number, the lower the impact of a possible evaluation error.

β’ Minimum subjectivity issue due to factors assessment performed after a questionnaire-based investigation.

β’ Possibility to choose which influence factors represent the best alternative to fit the specific system judging on system features. Thus, only the factors that actually influence the system performances are taken into consideration.

β’ Low complexity.

The main drawback of the AWM method is the equal weight that the influence factors have in the final equations (5.46) and (5.47).